How can I sue someone for an auto accident?

You sue them by filing a summons and complaint against them and having it served on them as required by your state's court rules. If the amount is under the limit for small claims court and the driver is in your county, you generally just file the forms in court and the court serves them while also assigning you a court date. If the amount is greater than the small claims limit or the driver is not local, you would have to sue in regular county court, which is more complicated and expensive, and you would then, after filing, have to serve the the summons and complaint yourself (generally by hiring a process server).
You can get instructions and forms from your court or the county/state court system generally: go to the court website first, though you can also go to the court clerk's office if necessary. Small claims cases are deliberately simple--they are intended for non-lawyers to be able to pursue them. A county court case can be difficult for a non-lawyer and has many procedural "pitfalls" which can trip you up--you'd be well advised to retain an attorney to help you.
To win, you will have to prove in court by a "preponderance of the evidence" (that it is more likely than not) that the other driver was at fault (e.g. driving carelessly or negligently) in causing the accident; only at-fault drivers are liable.
Bear in mind that suing, even suing and winning, still does not guaranty you will recover money. If they don't pay voluntarily after you win in court, you'll have to pursue collections efforts, which can be slow, difficult, and which cost money. And if they are insolvement and can't pay--and many people who don't have insurance don't have it because they are insolvent and can't afford it--you'll never get your money.